Electrostatic Correlations and the Polyelectrolyte Self Energy

We address the effects of chain connectivity on electrostatic fluctuations in polyelectrolyte solutions using a field-theoretic, renormalized Gaussian fluctuation (RGF) theory. As in simple electrolyte solutions (Z.-G. Wang, Phys. Rev. E. {\bf 81}, 021501 (2010)), the RGF provides a unified theory for electrostatic fluctuations, accounting for both dielectric and charge correlation effects in terms of the self-energy. Unlike simple ions, the polyelectrolyte self energy depends intimately on the chain conformation, and our theory naturally provides a self-consistent determination of the response of intramolecular chain structure to polyelectrolyte and salt concentrations. The theory captures the expected scaling behavior of chain size from the dilute to semi-dilute regimes; by properly accounting for chain structure the theory provides improved estimates of the self energy in dilute solution and correctly predicts the eventual $N$-independence of the critical temperature and concentration of salt-free solutions of flexible polyelectrolytes. We show that the self energy can be interpreted in terms of an infinite-dilution energy $\mu^\text{el}_{m,0}$ and a finite concentration correlation correction $\mu^\text{corr}$ which tends to cancel out the former with increasing concentration.